JEE Advance - Mathematics (1981 - No. 12)
Let the complex number $${{z_1}}$$, $${{z_2}}$$ and $${{z_3}}$$ be the vertices of an equilateral triangle. Let $${{z_0}}$$ be the circumcentre of the triangle. Then prove that $$z_1^2 + z_2^2 + z_3^2 = 3z_0^2$$.
The given equation holds true if and only if the triangle is equilateral and $$z_0$$ is its circumcenter.
The equation $$z_1^2 + z_2^2 + z_3^2 = 3z_0^2$$ is always true for any three complex numbers and their circumcenter.
The equation only holds if $$z_1$$, $$z_2$$, and $$z_3$$ are real numbers.
The equation is valid only when the origin is at the circumcenter.
None of the above